Moments of the Riemann Zeta-function

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. Recent work by Conrey et al [2] gives a more precise conjecture, identifying lower order terms in an asymptotic expansion for Mk(T ). Despite many attempts, asymptotic formulae for Mk(T ) have been established only for k = 1 (due to Hardy and Littlewood, see [19]) and k = 2 (due to Ingham, see [19]). However we do have the lower bound Mk(T ) ≫k T (logT ) 2 . This was established by Ramachandra [12] for positive integers 2k, by Heath-Brown [5] for all positive rational numbers k, and assuming the truth of the Riemann Hypothesis by Ramachandra [11] for all positive real numbers k. See also the elegant note [1] giving such a bound assuming RH, and [17] for the best known constants implicit in these lower bounds. Analogous conjectures exist (see [2], [4], [10]) for moments of central values of L-functions in families, and in many cases lower bounds of the conjectured order are known (see [14] and [15]). Here we study the problem of obtaining upper bounds for Mk(T ). When 0 ≤ k ≤ 2, Ramachandra ([12], [13]) and Heath-Brown ([5], [6]) showed, assuming RH, that Mk(T ) ≪ T (logT ) 2 . The Lindelöf Hypothesis is equivalent to the estimate Mk(T ) ≪k,ǫ T 1+ǫ for all natural numbers k. Thus, for k larger than 2, it seems difficult to make unconditional progress on bounding Mk(T ). If we assume RH, then a classical bound of Littlewood (see [19]) gives that (for t ≥ 10 and some positive constant C)